I'm looking for an introduction to the non-arithmetic aspects of the moduli of elliptic curves. I'd particularly like one that discusses the $H^1$ local system on the moduli space (whether it's $Y(1)$ or $Y(2)$ or whatever doesn't matter) from the Betti point of view ($SL_2(Z)$, representations of the fundamental group, etc) and the de Rham point of view (Picard-Fuchs equation, hypergeometric functions, etc). This is for a student who has taken classes in algebraic topology and complex analysis and who is just learning algebraic geometry, so I would prefer something that's as down to earth as possible, not a full-blown sheaf-theoretic treatment (i.e. with $R^1f_*$, D-modules, etc). This is a beautiful classical subject, so I can't believe there aren't really great expositions out there, but I can't think of even one!

[Edit 2010/01/21: Thanks to every who suggested references below. I'll probably suggest Clemens's book and Hain's notes. But after looking at them, I realize that I'd really like something even more basic, with no algebraic geometry (cubic curves) and no holomorphic geometry (Riemann surfaces). I just want the moduli spaces of homothety classes of lattices in C (maybe plus some level structure), viewed first as a topological space and later as a differentiable manifold, together with the H^1 local systems, viewed first as a representation of the fundamental group and later as a vector bundle with connection. Probably this is so easy that no one ever bothered to write it down, but on the chance that that's not the case, consider this a renewed request in more precise form.]

I spent the last year or so working on the project James outlined above. I ended up getting fairly sidetracked trying to figure out precisely what is meant by a moduli space, and more generally by a family of elliptic curves. The most helpful text I found for this was Kodaira's Complex Manifolds and Deformation of Complex Structures.

Although I didn't end up doing too much work with it, some useful references that discuss the main problem of the $H^1$ local system on the moduli space are the introduction to Period Mappings and Period Domains by Carlson et al, Yoshida's Hypergeometric Functions, My Love (and also other books/papers by Yoshida), and Holzapfel's Geometry and Arithmetic Around Euler Partial Differential Equations. The last book mainly deals with the genus two case, but does have a few pages on the genus one case. None of these references are perfect for the topic, at least coming from my level, but they all helped a fair bit.

I'm not sure that there are any good texts doing this out there. It is something I typically try to have my students learn, but it is always a little difficult because of the lack of a text. Most of the material is covered (but not at the level you want) in the first few pages of Deligne's paper on $\ell$-adic Galois representations.

There he covers the upper half-plane and its quotients as analytic moduli spaces of elliptic curves, does the local system of $H^1$s, and discusses the de Rham theory of these local systems and the relation to modular forms. He doesn't derive the Picard-Fuchs equation.
(I do have a short note somewhere in my files where I derive Picard-Fuchs from first principles, which I'd be happy to send you, if you want and if you think it would be useful.)

Deligne's treatment is very beautiful, as you would expect, but is fairly terse and abbreviated (as you might also expect). Your student would certainly need to work closely with you to find their way through it.

No, I mean Deligne's 1969 (I think) Bourbaki seminar on modular forms and $\ell$-adic representations. I think that this particular Bourbaki seminar is not indexed on MathSciNet (I don't know why; it seems fairly unique in this regard), but it is in Lecture Notes in Math 179. It is the paper in which Deligne introduced the Galois representations attached to higher weight modular forms, thus answering a question of Serre about their existence, and ushering in a whole new direction of research in number theory.
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EmertonJan 21 '10 at 13:33